%
\newcommand{\where}{\;{\bf where}\;}
\newcommand{\ssort}[1]{\;{\bf sort}_{#1}}
\newcommand{\llet}{\;{\bf let}}
\newcommand{\iif}{\;{\bf if}}
\newcommand{\iifnot}{\;{\bf ifnot}}
\newcommand{\qosparA}{{\bf Assumptions \; QoS \; parameters:}}
\newcommand{\qosparG}{{\bf Guarantees \; QoS \; parameters:}}
%\newcommand{\prog}[1]{{\footnotesize \textsf{#1}\normalsize}}
\newcommand{\prog}[1]{{\small\textsf{#1}\normalsize}}
\newcommand{\bprog}[1]{{\small\textbf{#1}\normalsize}}
\newcommand{\pprog}[1]{{\small\texttt{#1}\normalsize}}
\newcommand{\bemph}[1]{{\small\textbf{#1}\normalsize}}
%\newcommand{\pprog}[1]{{\small{\tt #1}\normalsize}}
\newcommand{\qual}[1]{{:\in_{#1}}}
\newcommand{\down}[1]{\left\lceil{#1}\right\rceil}

\newcommand{\causes}{\preceq}
\newcommand{\strictcauses}{\prec}


%\newcommand{\nnet}{\mathcal{N}}

\newcommand{\aleq}{\leq_\assDom}
\newcommand{\ageq}{\geq_\assDom}
\newcommand{\sal}{<_\assDom^s}
\newcommand{\sag}{>_\assDom^s}
\newcommand{\saleq}{\aleq^s}
\newcommand{\sageq}{\ageq^s}
\newcommand{\sgeq}{\geq^s}
\newcommand{\sleq}{\leq^s}
\newcommand{\compet}{\lhd}
\newcommand{\sgeqcompet}{\geqcompet^s}
\newcommand{\sleqcompet}{\leqcompet^s}
\newcommand{\lcompet}{<}
\newcommand{\leqcompet}{\leq}
\newcommand{\gcompet}{>}
\newcommand{\geqcompet}{\geq}
\newcommand{\dq}{"}

\newcommand{\qos}{\xi}
\newcommand{\aqos}{\alpha}
\newcommand{\bqos}{\beta}
\newcommand{\QOS}{Q}
\newcommand{\Qinit}{\QOS_{\rm init}}
\newcommand{\GQOS}{G}
\newcommand{\GQinit}{\GQOS_{\rm init}}
\newcommand{\AQOS}{A}
\newcommand{\AQinit}{A_{\rm orch}}
\newcommand{\ass}{\alpha}
\newcommand{\assDom}{\mathbb{A}}
\newcommand{\Dom}{\mathbb{D}}
\newcommand{\ASS}{A}
\newcommand{\Ainit}{\ASS_{\rm init}}
\newcommand{\zero}{0}
\newcommand{\orc}{Orc}
%\newcommand{\udoma}{\mathbb{D}^A}
%\newcommand{\udomg}{\mathbb{D}^G}
\newcommand{\udoma}{\mathbb{A}}
\newcommand{\udomg}{\mathbb{Q}}
\newcommand{\nnet}{\net}
\newcommand{\subnet}{S}
\newcommand{\colorset}{\Sigma}
\newcommand{\placeset}{\mathcal{P}}
\newcommand{\transset}{\mathcal{T}}
\newcommand{\flowset}{\mathcal{F}}
\newcommand{\proba}{\mathbf{P}}
\newcommand{\Probas}{\mathbb{P}}
%\newcommand{\datefamilies}{\mathbb{T}}
%\newcommand{\dateinit}{\mathbb{T}_{\rm init}}
\newcommand{\datefamilies}{\mathbb{Q}}
\newcommand{\dateinit}{\mathbb{Q}_{\rm init}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\readset}{\mathcal{R}}
\newcommand{\colormap}{\mathcal{C}}
\newcommand{\guardfn}{\mathcal{G}}
\newcommand{\expressionfn}{\mathcal{E}}
\newcommand{\initialmark}{\mathcal{I}}
\newcommand{\ie}{\emph{i.e}}
\newcommand{\eg}{\emph{e.g}}
\newcommand{\preset}[1]{{{}^{\bullet\!}{#1}}}
\newcommand{\postset}[1]{#1^\bullet}
\newcommand{\after}[1]{\widehat{#1}}
%\newcommand{\context}[1]{{\underline{#1}}}
%\newcommand{\presetcontext}[1]{{{}^\bullet\underline{#1}}}
\newcommand{\context}[1]{{{}^{\circ\!}{#1}}}
\newcommand{\presetcontext}[1]{{{}^{\underline{\bullet\!}\,}{#1}}}
%\newcommand{\cone}[1]{\lbrack{#1}\rbrack}
%\newcommand{\cone}[1]{\left\lceil#1\right\rceil}
\newcommand{\cone}[1]{\lceil#1\rceil}
\newcommand{\asconfst}{\natural}
\newcommand{\asconf}{\nearrow}
\newcommand{\lasconf}{\nwarrow}
\newcommand{\lasconfconf}{\;_{_\#\!\!\!\!\!\!}\nwarrow}
\newcommand{\asconfmin}{\asconf_{\!\!\!\!_{\#{m}}}}
\newcommand{\asconfconf}{\asconf_{\!\!\!\!_\#}}
\newcommand{\config}{\kappa}
\newcommand{\mclique}[1]{mcc(#1)}
\newcommand{\cluster}{\mathbf{c}}
%\newcommand{\maxconfig}{\overline{\kappa}}
%\newcommand{\barconfig}{\bar{\kappa}}
\newcommand{\maxconfig}{\kappa}
\newcommand{\barconfig}{\kappa}
\newcommand{\minTrans}[1]{minTrans(#1)}
\newcommand{\maxPlaces}{{\it maxPlaces}}
\newcommand{\Confset}{\mathcal{V}}
\newcommand{\enabledbc}{\delta}

%\newcommand{\cocnets}{\emph{COC}-nets }
%\newcommand{\cocnet}{\emph{COC}-net }

\newcommand{\cocnets}{\emph{CO}-nets }
\newcommand{\cocnet}{\emph{CO}-net }

\newcommand{\Tfam}{\Delta }

\newcommand{\type}[1]{\mathrm{Type}(#1)}
%%NEW COMMANDS ALBERT
%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\daemon}{\omega}
\newcommand{\Daemons}{\Omega}
\newcommand{\orchnet}{OrchNet}
%\newcommand{\preorchnet}{pre-OrchNet}
\newcommand{\wfnet}{WFnet}
%\newcommand{\prewfnet}{pre-WFnet}
\newcommand{\Orchnet}{\mathcal{N}}
%\newcommand{\Preorchnet}{\mathbb{N}}
\newcommand{\WFnet}{\mathcal{W}}
%\newcommand{\PreWFnet}{\mathbb{W}}
\newcommand{\net}{N}
\newcommand{\mnet}{M}
\newcommand{\prefix}{M}
\newcommand{\wnet}{W}
\newcommand{\Tinit}{T_{{\rm init}}}
\newcommand{\Finit}{F_{{\rm init}}}
\newcommand{\event}{t}
\newcommand{\data}{\nu}
\newcommand{\Data}{V}
\newcommand{\bcell}{x}
\newcommand{\Bcells}{\Xi}
%\newcommand{\config}{v}
\newcommand{\concurrent}{\|}
\newcommand{\conflict}{\#}
%\newcommand{\Configs}[1]{\mathcal{V}_{#1}}
%\newcommand{\MaxConfigs}[1]{\overline{\mathcal{V}}_{#1}}
\newcommand{\closure}[1]{\cone{#1}}
%\newcommand{\closure}[1]{\left\lceil#1\right\rceil}
\newcommand{\future}[2]{{#1}^{#2}}


%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\af}[1]{\
\mbox{{\boldmath\scriptsize{$>$}}$#1${\boldmath\scriptsize{$>$}}}\ }
\newcommand{\aft}{\mbox{\boldmath{\scriptsize {$\: \gg\: $}}}}
\newcommand{\aftsmall}{\mbox{\boldmath{\small {$\: \gg\: $}}}}
\newcommand{\pull}[1]{\
\mbox{{\boldmath\scriptsize{$<$}}$#1${\boldmath\scriptsize{$<$}}}\ }
\newcommand{\QoSpull}[2]{\
\mbox{{\boldmath\scriptsize{$<$}}$#1${\boldmath\scriptsize{$<_{#2}$}}}\ }
\newcommand{\dfn}{\mathrm{\underline{\Delta}}}
\newcommand{\RTimer}{{\bf RTimer}}

\newcommand{\pre}[1]{\downarrow\!\!#1}
\newcommand{\tr}[2]{[\![#1]\!]_{#2}}

%\newtheorem{assumption}{Assumption}
%\newtheorem{algorithm}{Procedure}

%\definecolor{light-gray}{gray}{0.55}
